Advertisements
Advertisements
प्रश्न
If α and β are the complex cube roots of unity, show that α2 + β2 + αβ = 0.
Advertisements
उत्तर
α and β are the complex cube roots of unity.
∴ α = `(-1 + i sqrt (3))/2 and beta = (-1 - i sqrt (3))/2`
∴ αβ = `((- 1 + i sqrt (3))/2)((-1 - i sqrt(3))/2)`
= `((-1)^2 - (i sqrt (3))^2)/4`
= `(1 - (-1)(3))/4` ...[∵ i2 = – 1]
= `(1 + 3)/4`
∴ αβ = 1
Also, α + β = `(-1 + i sqrt (3))/2 + (-1 - i sqrt(3))/2`
= `(-1 + i sqrt(3) - 1 - i sqrt (3))/2`
= `(-2)/2`
∴ α + β = – 1
L.H.S. = α2 + β2 + αβ
= α2 + 2αβ + β2 + αβ – 2αβ ...[Adding and subtracting 2αβ]
= (α2 + 2αβ + β2) – αβ
= (α + β)2 – αβ
= (– 1)2 – 1
= 1 – 1
= 0 = R.H.S.
APPEARS IN
संबंधित प्रश्न
If ω is a complex cube root of unity, then prove the following: (ω2 + ω - 1)3 = – 8
If ω is a complex cube root of unity, then prove the following: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω18
If ω is a complex cube root of unity, show that (2 − ω)(2 − ω2) = 7
If ω is a complex cube root of unity, show that (3 + 3ω + 5ω2)6 − (2 + 6ω + 2ω2)3 = 0
If ω is a complex cube root of unity, find the value of (1 + ω)(1 + ω2)(1 + ω4)(1 + ω8)
Find the equation in cartesian coordinates of the locus of z if |z − 5 + 6i| = 5
Find the equation in cartesian coordinates of the locus of z if |z + 8| = |z – 4|
Let α be a root of the equation 1 + x2 + x4 = 0. Then the value of α1011 + α2022 – α3033 is equal to ______.
If 1, α1, α2, ...... αn–1 are the roots of unity, then (1 + α1)(1 + α2) ...... (1 + αn–1) is equal to (when n is even) ______.
If w is a complex cube root of unity, show that
`((a + bw + cw^2)) /( c + aw + bw^2 )= w^2`
If w is a complex cube-root of unity, then prove the following:
(ω2 + ω − 1)3 = −8
If w is a complex cube root of unity, show that `((a + bw + cw^2))/(c+aw+bw^2) = w^2`
If ω is a complex cube root of unity, then prove the following.
(ω2 + ω −1)3 = −8
If ω is a complex cube-root of unity, then prove the following :
(ω2 + ω − 1)3 = − 8
Find the value of `sqrt(-3) xx sqrt(-6)`.
If w is a complex cube-root of unity, then prove the following
(w2 + w - 1)3 = - 8
